Vortex dynamics of a sphere wake in proximity to a wall

Toward getting the vortex dynamics characteristics and wake structure of a sphere in proximity to a wall, the effect of a proximal flat plate on the wake of a stationary sphere is investigated via direct numerical simulation. The vortex shedding process and the significant variation of the wake structure are described in detail. The drag coefficient reduces and the wake structure of the sphere becomes complex due to the combined effect of the wake flow and the wall. A jet flow forms between the sphere and the flat plate, which suppresses the vortex separation on the bottom of the sphere. The asymmetric distributions of the coherent structures and the recirculation region behind the sphere are discussed. Besides vortex shedding patterns, the time-averaged velocity distribution, vortex dynamics, distribution regularities of turbulent kinetic energy and enstrophy are investigated.     

Flow past a sphere in density-stratified fluid

Stratified flow past a three-dimensional obstacle such as a sphere has been a long-lasting subject of geophysical, environmental and engineering fluid dynamics. In order to investigate the effect of the stratification on the near wake, in particular, the unsteady vortex formation behind a sphere, numerical simulations of stratified flows past a sphere are conducted. The time-dependent Navier–Stokes equations are solved using a three-dimensional finite element method and a modified explicit time integration scheme. Laminar flow regime is considered, and linear stratification of density is assumed under Boussinesq approximation. The effects of stratification is implemented by density transport without diffusion. The computed results include the characteristics of the near wake as well as the effects of stratification on the separation angle. Under increased stratification, the separation on the sphere is suppressed and the wake structure behind the sphere becomes planar, resembling that behind a vertical cylinder. With further increase in stratification, the wake becomes unsteady, and consists of planar vortex shedding similar to von Karman vortex streets.     

The flow past a freely rotating sphere

We consider the flow past a sphere held at a fixed position in a uniform incoming flow but free to rotate around a transverse axis. A steady pitchfork bifurcation is reported to take place at a threshold \(Re^\mathrm{OS}=206\) leading to a state with zero torque but nonzero lift. Numerical simulations allow to characterize this state up to \(Re\approx 270\) and confirm that it substantially differs from the steady-state solution which exists in the wake of a fixed, non-rotating sphere beyond the threshold \(Re^\mathrm{SS}=212\). A weakly nonlinear analysis is carried out and is shown to successfully reproduce the results and to give substantial improvement over a previous analysis (Fabre et al. in J Fluid Mech 707:24–36, 2012). The connection between the present problem and that of a sphere in free fall following an oblique, steady (OS) path is also discussed.     

Conjugate heat transfer in the unbounded flow of a viscoelastic fluid past a sphere

This work addresses the conjugate heat transfer of a simplified PTT fluid flowing past an unbounded sphere in the Stokes regime (Re = 0.01). The problem is numerically solved with the finite-volume method assuming axisymmetry, absence of natural convection and constant physical properties. The sphere generates heat at a constant and uniform rate, and the analysis is conducted for a range of Deborah (0 ≤ De ≤ 100), Prandtl (10⁰ ≤ Pr ≤ 10⁵) and Brinkman (0 ≤ Br ≤ 100) numbers, in the presence or absence of thermal contact resistance at the solid–fluid interface and for different conductivity ratios (0.1 ≤ κ ≤ 10). The drag coefficient shows a monotonic decrease with De, whereas the normalized stresses on the sphere surface and in the wake first increase and then decrease with De. A negative wake was observed for the two solvent viscosity ratios tested (β = 0.1 and 0.5), being more intense for the more elastic fluid. In the absence of viscous dissipation, the average Nusselt number starts to decrease with De after an initial increase. Heat transfer enhancement relative to an equivalent Newtonian fluid was observed for the whole range of conditions tested. The dimensionless temperature of the sphere decreases and becomes more homogeneous when its thermal conductivity increases in relation to the conductivity of the fluid, although small changes are observed in the Nusselt number. The thermal contact resistance at the interface increases the average temperature of the sphere, without affecting significantly the shape of the temperature profiles inside the sphere. When viscous dissipation is considered, significant changes are observed in the heat transfer process as Br increases. Overall, a simplified PTT fluid can moderately enhance heat transfer compared to a Newtonian fluid, but increasing De does not necessarily improve heat exchange.     

Dynamic interaction of fixed dual spheres for several configurations and inflow conditions

The changes in force characteristics as well as the shedding patterns for various dual sphere configurations are studied. The Reynolds numbers considered are 300, 600 and two different inflow conditions are used: steady and pulsating. The sphere formations are defined by the separation distance D₀ between the spheres and the angle between the line connecting the centres of the spheres and the main flow direction, γ. The position of one of the spheres is varied in the range 0°–90° using a 15° increment. Two separation distances are studied; 1.5D and 3D. The method used for the simulations is the Volume of Solid (VOS) approach, a method based on Volume of Fluid (VOF). A major conclusion from this work is that the sphere interaction alters the wake dynamics by obstructing the vortex shedding (generating a steady wake or a wake with lower Strouhal number) and by changing the direction of the lift force so that it in most cases is directed in the plane containing the sphere centres. The results also show that changing the inflow condition gives the same relative change in drag and lift as for a single sphere. The drag is substantially reduced by placing the sphere downstream in a tandem arrangement and slightly increased in a side-by-side arrangement. However, the effect is decreased by increasing separation distance and increasing Reynolds number.     

Numerical Investigation of the Flow Past a Rotating Golf Ball and Its Comparison with a Rotating Smooth Sphere

Large-eddy simulations are conducted for a rotating golf ball and a rotating smooth sphere at a constant rotational speed at the subcritical, critical and supercritical Reynolds numbers. A negative lift force is generated in the critical regime for both models, whereas positive lift forces are generated in the subcritical and supercritical regimes. Detailed analysis on the flow separations on different sides of the models reveals the mechanism of the negative Magnus effect. Further investigation of the unsteady aerodynamics reveals the effect of rotating motion on the development of lateral forces and wake flow structures. It is found that the rotating motion helps to stabilize the resultant lateral forces for both models especially in the supercritical regime.     

Flow of a particulate suspension in the wake of a circular cylinder

Measurements are reported for the average local particulate velocity and concentration distributions in the wake of a cylinder immersed in a stream containing a polydisperse aerosol. The wake centerline defects and transverse distributions were determined for both parameters. It was found that the particulate centerline defect persists a considerable distance downstream of the cylinder before fully developed conditions are satisfied. Transverse particulates and gaseous velocity distributions assume a Gaussian profile at the same point downstream. The charge-to-mass ratio throughout the wake region was equal to the free stream value for all experimental conditions and was of such a magnitude to permit the electrostatic effects to be neglected in the governing equations. Gaseous and particulate transport properties were identified in the wake.     

Numerical simulation on steady flow around and through a porous sphere

The flow around and through a porous sphere is investigated numerically. It is found that the recirculating wake existing downstream of the porous sphere is either completely detached from or penetrates it. Specially, in a certain range of Darcy number, the wake may initially increase in size with an increase in Reynolds number but then decrease in size and eventually disappear when the Reynolds number is sufficiently large. For a small Darcy number, the critical Reynolds number for the onset of recirculating wake may be significantly smaller than that of a solid sphere. The present study suggests that the surface curvature has an important effect on the initial position of the onset of recirculating wake.     

Effect of ventilation on the flowfield around a sphere

The flowfield around a sphere with and without ventilation was investigated in a wind tunnel over a range of Reynolds numbers in an incompressible flow. At supercritical Re, the pressure drag of a sphere can be nearly nullified by venting only 2% of the frontal area of the sphere to the base through a smooth internal duct. The drag reduction is achieved by increased pressures in the separated flow region close to the base. At high Re, the vent flow breaks through the near wake and brings about symmetry in the global flowfield. When the internal shear is increased by using a rough internal duct, the base pressure is unchanged, but the external flow is accelerated to velocities beyond that achieved by the potential flow around the basic sphere. The findings can be explained by a flow model in which the near wake is aerodynamically streamlined by a pair of counterrotating vortex rings at the base. A roughness element can be made to partially destroy the vortex system at the base and result in a steady asymmetric wake. A 1.2 mm diameter wire placed at 70° was found to overtrip the boundary layer and completely destroy the vortex system. Simultaneously, the turbulent separation is advanced and the drag increased.At subcritical Re, ventilation marginally increases static pressures all over the surface. Since the large pressure differential between the windward and leeward sides is not reduced, the internal flow has a rapid acceleration to a velocity close to that of the free stream. The reverse flow associated with the near wake forces the vent flow to rest within itself and the wake profile is unchanged. The main features of subcritical flow around the basic sphere are retained in spite of ventilation. The upstream effects of ventilation are greater for subcritical flow than for supercritical flow.The work reported was carried out under a study grant from the German Academic Exchange Service (DAAD) in Bonn. The authors wish to thank the Director of DAAD in Bonn for the same. Thanks are due to Dr. F. R. Grosche and colleagues at DLR in Göttingen who assisted in the design, fabrication and wind tunnel testing of the sphere model. Thanks are also due to Prof. D. G. Mabey, visiting Professor, Imperial College, London for useful discussions. The many useful discussions with the research advisors of the first author viz., Dr. P. R. Viswanath of National Aerospace Laboratories and Prof. A. Prabhu of Indian Institute of Science, Bangalore are acknowledged with thanks. The support given by the Head, Experimental Aerodynamics Division, National Aerospace Laboratories is thankfully acknowledged.     

Experimental study of the mean wake of a tidal stream rotor in a shallow turbulent flow

The mean wake of a three-bladed horizontal axis tidal stream turbine operating at maximum power coefficient has been investigated experimentally in a wide flume with width 11 times the depth, providing minimal restriction to transverse wake development and behaviour of large-scale horizontal turbulence structures. This is an important first stage for understanding wake interaction in turbine arrays and hence large-scale power generation. The rotor diameter has a typical value of 60% of the depth and the thrust coefficient is representative of a full-scale turbine. The shear layers originating from the rotor tip circumference show classic linear expansion downstream, with the rate of a plane shear layer vertically and 1.5 times that horizontally. These shear layers merge by around 2.5 diameters downstream forming a self-similar two-dimensional wake beyond eight diameters downstream with a virtual origin at two diameters downstream of the rotor plane. The spreading rate is somewhat less than that for solid bodies. The detailed velocity measurements made in the near wake show rotation and vorticity similar to that measured previously for wind and marine turbines although with asymmetry associated with bed and surface proximity. The longitudinal circulation in a transverse plane is conserved at about 1% of the swept circulation from the blade tip within two diameters downstream, the extent of detailed measurement. Turbines are usually designed using blade element momentum theory in which velocities at the rotor plane are characterised by axial and tangential induction factors and it is now possible to see how this idealisation relates to actual velocities. The axial induction factor corresponds to velocity deficits at 0.4–0.8 radii from the rotor axis across the near wake while the tangential induction factor at the rotor plane corresponds to velocities at 0.4–0.6 radii between 1–2 diameters downstream, indicating some general correspondence. For the two-dimensional self-similar far wake the two parameters defining the centreline velocity deficit and the transverse velocity profiles are likely to be insensitive to Reynolds number in turbulent conditions.     

Controlled oscillations of a cylinder: a new wake state

The wake states resulting from the controlled oscillation of a cylinder transverse to the free stream are presented. A new wake state is revealed by instantaneous measurements of the total and vortex lift phases, and the phase-referenced, quantitative wake structure. This “intermediate wake state” occurs at oscillation frequencies between the previously observed low- and high-frequency states. It cannot be deduced from classical, time-averaged representations of the loading on the cylinder.     

Numerical study of transient laminar natural convection heat transfer over a sphere subjected to a constant heat flux

Transient laminar natural convection over a sphere which is subjected to a constant heat flux has been studied numerically for high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and a wide range of Prandtl numbers (Pr = 0.02, 0.7, 7, and 100). A plume with a mushroom-shaped cap forms above the sphere and drifts upward continuously with time. The size and the level of temperature of the transient cap and plume stem decrease with increasing Gr and Pr. Flow separation and an associated vortex may appear in the wake of the sphere depending on the magnitude of Gr and Pr. A recirculation vortex which appears and grows until “steady state” is attained was found only for the very high Grashof numbers (10⁵ ≤ Gr ≤ 10⁹) and the lowest Prandtl number considered (Pr = 0.02). The appearance and subsequent disappearance of a vortex was observed for Gr = 10⁹ and Pr = 0.7. Over the lower hemisphere, the thickness of both the hydrodynamic (δ_H) and the thermal (δ_T) boundary layers remain nearly constant and the sphere surface is nearly isothermal. The surface temperature presents a local maximum in the wake of the sphere whenever a vortex is established in the wake of the sphere. The surface pressure recovery in the wake of the sphere increases with decreasing Pr and with increasing Gr. For very small Pr, unlike forced convection, the ratio δ_T/δ_H remains close to unity. The results are in good agreement with experimental data and in excellent agreement with numerical results available in the literature. A correlation has also been presented for the overall Nusselt number as a function of Gr and Pr.     

高超声速尾迹流场稳定性数值研究

通过数值模拟, 对高超声速尾迹流场进行了研究, 对其尾迹流动的失稳过程进行了分析.选取计算模型为圆球,Ma= 6.0, Re = 1.71\times 10^6(Re以球头半径为参考长度). 通过数值模拟,首先得到的流动是稳定解,在底部发展出一个主分离区和一个二次分离区,流动是轴对称状态. 不添加任何扰动继续进行计算,发现底部流场缓慢发展出微弱的非定常流动. 随后,该现象继续发展,出现明显的结构失稳,得到了无量纲周期为12.0的周期解. 给出了高超声速圆球绕流尾迹结构的周期性演化过程,对其涡系结构的演化及奇点特征进行了分析. 研究表明该数值模拟方法可用于底部流动稳定性问题的研究,同时证实了高超声速底部流动也存在流动不稳定性.      

An experimental investigation of negative wakes behind spheres settling in a shear-thinning viscoelastic fluid

We present detailed experimental results examining “negative wakes” behind spheres settling along the centerline of a tube containing a viscoelastic aqueous polyacrylamide solution. Negative wakes are found for all Deborah numbers (2.43≤De(˙γ)≤8.75) and sphere-to-tube aspect ratios (0.060≤a/R≤0.396) examined. The wake structures are investigated using laser-Doppler velocimetry (LDV) to examine the centerline fluid velocity around the sphere and digital particle image velocimetry (DPIV) for full-field velocity profiles. For a fixed aspect ratio, the magnitude of the most negative velocity, U _min , in the wake is seen to increase with increasing De. Additionally, as the Deborah number becomes larger, the location of this minimum velocity shifts farther downstream. When normalized with the sphere radius and the steady state velocity of the sphere, the axial velocity profiles become self-similar to the point of the minimum velocity. Beyond this point, the wake structure varies weakly with aspect ratio and De, and it extends more than 20 radii downstream. Inertial effects at high Reynolds numbers are observed to shift the entire negative wake farther downstream. Using DPIV to investigate the transient kinematic response of the fluid to the initial acceleration of the sphere from rest, it is seen that the wake develops from the nonlinear fluid response at large strains. Measurements of the transient uniaxial extensional viscosity of this weakly strain-hardening fluid using a filament stretching rheometer show that the existence of a negative wake is consistent with theoretical arguments based on the opposing roles of extensional stresses and shearing stresses in the wake of the sphere. Received: 10 November 1997 Accepted: 1 May 1998     

Determination of real-time predictors of the wind turbine wake meandering

The present work proposes an experimental methodology to characterize the unsteady properties of a wind turbine wake, called meandering, and particularly its ability to follow the large-scale motions induced by large turbulent eddies contained in the approach flow. The measurements were made in an atmospheric boundary layer wind tunnel. The wind turbine model is based on the actuator disc concept. One part of the work has been dedicated to the development of a methodology for horizontal wake tracking by mean of a transverse hot wire rake, whose dynamic response is adequate for spectral analysis. Spectral coherence analysis shows that the horizontal position of the wake correlates well with the upstream transverse velocity, especially for wavelength larger than three times the diameter of the disc but less so for smaller scales. Therefore, it is concluded that the wake is actually a rather passive tracer of the large surrounding turbulent structures. The influence of the rotor size and downstream distance on the wake meandering is studied. The fluctuations of the lateral force and the yawing torque affecting the wind turbine model are also measured and correlated with the wake meandering. Two approach flow configurations are then tested: an undisturbed incoming flow (modelled atmospheric boundary layer) and a disturbed incoming flow, with a wind turbine model located upstream. Results showed that the meandering process is amplified by the presence of the upstream wake. It is shown that the coherence between the lateral force fluctuations and the horizontal wake position is significant up to length scales larger than twice the wind turbine model diameter. This leads to the conclusion that the lateral force is a better candidate than the upstream transverse velocity to predict in real time the meandering process, for either undisturbed (wake free) or disturbed incoming atmospheric flows.     

A FLOW VISUALIZATION STUDY ON FEEDBACK CONTROL OF VORTEX SHEDDING FROM A CIRCULAR CYLINDER

This note presents flow visualization results to show the response of wake flows behind a cylinder to the feedback suppression and excitation. The experiments were conducted in a water channel and the feedback perturbations were introduced into the wake by oscillating the cylinder transverse to the oncoming flow. The visualization photographs directly illustrated the wake flows under the feedback suppression and excitation at Reynolds numbers up to 25% above the natural onset Reynolds number for vortex shedding.     

Transverse motions of a single sphere in hanging or swing configurations under a longitudinal flow

As a result of the reporting of casual observations of the oscillation or rotation of the beacons in transmission line guard cables, some attention has been paid to the stability of the guard cables with beacons.The relatively more frequent observation of these motions has been explained in recent papers dealing with the elastic part of the problem as a consequence of the increasing number of resonant frequencies (one for each additional beacon) that can be excited by appropriate aerodynamic loads. But a model that could explain the aerodynamic forces that can give rise to this motion is still lacking.In this paper we consider the transverse motions of a single sphere in two simplified configurations, (1) hanging (tethered at one point), and (2) swing (tethered at two points) under a longitudinal flow, performing small amplitude swinging oscillations or circular-orbit autorotation about an axis parallel to the main flow direction. The dynamic model here presented is based on the motion equations, which also include a model for the aerodynamic lift and drag forces on the sphere in transverse motion, which considers the effect of changes of flow around the sphere due to the cable interference. These forces are contained in the symmetry plane of the flow relative to the sphere, and, when projected on the lateral direction, give rise to a lateral force, which can explain the existence of the azimuthal motion even at a large reduced velocity, outside the vortex induced vibration (VIV) range The conditions for stable small oscillation motion and circular-orbit autorotation of a sphere in a swing configuration are given.The results for the aerodynamic loads in transverse motion have also been applied to the case of a circular-orbit autorotation of a hanging sphere (spherical pendulum) under a vertical flow. The angular rotation speed and the orbit radius (or cable angle) have been determined as a function of aerodynamic coefficients and configuration parameters.     

Hydrodynamic forces acting on a rigid fixed sphere in early transitional regimes

A spectral – spectral-element code is used to investigate the hydrodynamic forces acting on a fixed sphere placed in a uniform flow in the Reynolds number interval [10–320] covering the early stages of transition, i.e. the steady axisymmetric regime with detached flow, the steady non-axisymmetric and the unsteady periodic regimes of the sphere wake. The mentioned changes of regimes, shown by several authors to be related to a regular and a Hopf bifurcations in the wake, result in significant changes of hydrodynamic action of the flow on the sphere. In the present paper, we show that the loss of axisymmetry is accompanied not only by an onset of lift but also of a torque and we give accurate values of drag, lift and torque in the whole interval of investigated Reynolds numbers. Among other results show, moreover, that each bifurcation is accompanied also by a change of the trend of the drag versus Reynolds number dependence, the overall qualitative effect of instabilities being an increase of drag.     

Convective diffusion in a periodic array of spheres for small reynolds numbers

It is shown that in flow past a system of spheres of radius a situated at the nodes of a cubic lattice with the period b in the direction of one of the principal translations of the lattice under the condition (a/b) · · P^1/31 (P is the Péclet number, P1), the concentration of dissolved material absorbed by the sphere surfaces diminishes logarithmically at distances large compared with b, but small compared with L=Pb²/4a. At distances considerably larger than L, the decrease is described by an exponential law which coincides with the law of concentration decrease at distances much larger than b in the case of a spatially homogeneous distribution of the spheres. We consider the flow of an incompressible fluid with the velocity U past a system of spheres of radius a. We assume that the Reynolds number R=Ua/ (where , the kinematic viscosity coefficient, is much larger than unity). Dissolved in the fluid is a material of concentration c which is absorbed by the sphere surfaces. The diffusion coefficient D is assumed to be sufficiently small for the Péclet number P=Ua/D to be much larger than unity. The spheres are situated at the nodes of a cubic lattice with the period b. As will be shown below, it is necessary that P(a/b)³1. Under these assumptions the concentration varies in a thin (of the order aP^–1/3) diffusion layer near the surface of each sphere. A diffusion wake is formed behind each sphere. The transverse dimensions of this wake for a sufficiently widely spaced lattice (aP^1/3 b) exceed the effective thickness of the diffusion boundary layer, which enables us to reduce the problem of concentration absorption on the surface of the system of spheres to the problem considered by Levich [1] concerning the convective diffusion of a material of constant constant concentration flowing past a single sphere.Hasimoto [2] considers the solution of the Stokes equation describing the motion of a viscous fluid past an array of spheres situated at the nodes of a cubic lattice. However, he does not give an expression for the velocity field applicable near the surface of some single sphere which is necessary to the solution of the diffusion problem.In the method of Lamb [3] (§336) and Burgers [4], in dealing with the flow of a viscous stream past a single sphere, one considers the equation of motion in space, including the interior of the sphere, and not just the solution of the equation in the space outside the sphere with boundary conditions at the sphere surface. At the center of the sphere one places a concentrated force and a system of multipoles whose magnitude is chosen in such a way as to ensure fulfillment of the required boundary conditions.This idea of introducing an effective potential is used in [2] to find the velocity field of a fluid flowing past an array of spheres. We propose a treatment of the effective potential method somewhat different from that of [2].The authors are grateful to V. G. Levich and V. S. Krylov for their comments.